Fourier-Conjugate Models in the Corpuscular-Wave Dualism Concept

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1 International Journal of Adanced Reearch in Phyical Science (IJARPS) Volume, Iue 0, October 05, PP 6-30 ISSN (Print) & ISSN (Online) Fourier-Conjugate Model in the Corpucular-Wae Dualim Concept Yu.N. Dubnihche Intitute of Thermophyic, Siberian ranch, Ruian Academy of Science Nooibirk State Technical Unierity, Ruia T.Ya. Dubnihchea Nooibirk State Unierity of Economic and Management, Ruia Abtract: Decription of the wae-particle duality and the uncertainty relation baed on the Fourier conjugate mathematical model of the particle in the coordinate pace and in the frequency pace i dicued. The ignal recorded by an oberer in the coordinate pace and the impule pace are atified to the fundamental principle of uncertainty. It follow from the uncertainty relation for the extent of the ignal in the coordinate pace and for the width of it Fourier pectrum, a pecial cae of which are the relation of Heienberg uncertainty. Keyword: corpucular-wae dualim; de roglie wae, correlation ambiguitie, Fourier tranformation, uncertainty principle.. INTRODUCTION Corpucular-wae dualim i one of the concept ued to contruct the phyical pattern of the world, which are deeloped on the bai of ynthei of phyical image and analogie decribed in the mathematical language. A wa mentioned by de roglie, Poincare belieed that there exit an infinitely large number of logically equialent point and pattern of reality, while the reearcher chooe only one of them, baed excluiely on the iewpoint of conenience []. The traditional decription of corpucular-wae dualim i baed on quantization (ampling) of the energy of a moing corpucle (particle) by analogy with photon. The corpucular-wae dualim decription i uually baed on modeling of moing particle a wae packet and on the optical-mechanical analogy. Howeer, the deelopment of a imple and obiou model that explain how a moing particle acquire wae propertie i till an urgent tak. Thi work decribe a motiated attempt to contruct uch a model, baed on combining phyical and information analogie and contributing to undertanding of the wae nature of moing matter. The brilliant de roglie gue about wae propertie of moing particle wa baed on the Hamilton-Jacobi optical-mechanical analogy, quantization and preentation of light field by fluxe of light particle (photon) poeing wae propertie (Pauli, Eintein), and theoretical model implying the equialence of ma and energy (Poincare, Eintein). The idea of quantization i illutrated by a imple example of a harmonic ocillator. The motion of a harmonic ocillator in the phae plane (in the momentum-diplacement coordinate) i decribed by an elliptical phae trajectory. The area embraced by thi trajectory i equal to the ratio of the energy Е and frequency of ocillation, I E, which wa called the adiabatic inariant becaue it i conered at mall adiabatic changed in frequency. Thi mean that the ratio of the energy of ocillation of the harmonic ocillator to frequency i equal to the energy deriatie with repect to frequency or, if differential are replaced by mall increment, to the ratio of the correponding increment of energy and frequency: E de E I cont. d ARC Page 6

2 Yu.N. Dubnihche & T.Ya. Dubnihchea Thi equation ugget a poibility of quantization of the adiabatic inariant, which wa called the action in analytical mechanic. It i the action that i ampled rather than the energy, which i a continuou function of frequency. The harmonic ocillator whoe adiabatic inariant i ampled and whoe quantum of the action i equal to Planck contant h i a quantum ocillator. A pecific feature of the quantum ocillator i the fact that it phae trajectory cannot be a cloed cure in accordance with Heienberg correlation ambiguitie. The uncertainty of the phae trajectory poition on the phae plane i h, and the adiabatic inariant of the quantum ocillator i determined by the formula I n h. A the adiabatic inariant for the harmonic ocillator i much greater than Planck contant, it changing with frequency may be conidered a continuou and it phae trajectory may be conidered a cloed. The exitence of the quantum of the action h i a neceary, but not a ufficient condition for Heienberg correlation ambiguitie. It follow from corpucular-wae dualim, which, in turn, i baed on combining phyical image, analogie, and mathematical tranformation, including the Fourier tranformation. Let u conider a harmonic wae propagating in a coordinate pace and decribed by a periodic function of the form exp it kr. The phae of thi wae, t kr, i determined by the algebraic um of the time-dependent t and pace-dependent r component. The timedependent component of the phae i found a the product of the circular frequency and the time t. The circular frequency of the wae i determined a T, where T i the wae period. The pace-dependent component of the phae i the product of the wae ector k by the radiuector r, which decribe the patial poition of the point conidered. The abolute alue of the wae ector k i determined by the patial period of the wae, k k. y comparing T and k, we can ee that the frequency and the abolute alue of the wae ector k k hae imilar tructure. Therefore, the wae ector k ha the meaning of the patial frequency. In contrat to the frequency, the patial frequency k i a ector, which i determined by the projection k x, k y, and k z in the Carteian coordinate ytem. The wae i decribed in the coordinate ytem (t, x, y, z) in the coordinate pace and in the coordinate ytem (, k x, k y, k z ) in the frequency pace. Correpondingly, if a moing phyical object in the coordinate pace i defined by the mathematical model t,r, the Fourier pectrum,k correpond to thi model in the frequency pace. The frequency pace i equialent to the momentum pace becaue their coordinate differ by the factor h/ : p k and E. The motion of phyical object (corpucle, macrocopic body, or wae) can be decribed in the coordinate or frequency (momentum) pace. Thee mapping are related ia the Fourier tranformation and are conitent with the phyical reality. For intance, a flying ball can be decribed by it poition, elocity, and momentum; in the cae of it rotation, the call i alo decribed by the angular momentum. Certainly, here we mean mathematical model and ignal detected by the oberer rather than preentation and tranformation of object themele. In accordance with the claical definition of matter a the objectie reality gien to u in enation, ignal may be conidered a thi enation. The notion of the patial frequency and of the frequency and momentum pace are widely ued in cience and engineering. Example are optical information technologie and pectrocopy [, 3]. Let a moing material particle in the coordinate pace be decribed by the function r t, where r i the radiu-ector of the particle location, i the elocity ector of the particle, and t i the time. In the frequency pace, it i decribed by the Fourier pectrum,k, which i a function of the patial frequency k k (wae ector) and of the temporal frequency. The function t,k are related ia the Fourier tranform r and r t r texp ikrexp it drdt International Journal of Adanced Reearch in Phyical Science (IJARPS) Page 7

3 Fourier-Conjugate Model in the Corpucular-Wae Dualim Concept k i kt dt k exp, () where i the Dirac delta function. Here we ue the hift theorem and introduce the notation k. The function k i the Fourier pectrum,k of the function r t, which decribe the particle moing in the coordinate pace. A it follow from Eq. (), the Fourier pectrum of the ignal conidered a a moing particle in the frequency pace ha the form of a -function localized at the temporal frequency k and the patial frequency k. The direction of the corpucle elocity and the wae ector (patial frequency) k coincide, wherea the abolute alue of the wae ector k (wae number k) i determined by the ratio of the frequency to the elocity : k k The frequency equal to the product of the wae ector and the elocity ector of particle motion k, which i identical to the formula for the Doppler frequency hift and tetifie to the kinematic nature of the frequency. The elocity i determined by the relatie elocity of the corpucle and oberer coordinate ytem; for the wae induced by corpucle, it i determined by the group elocity. The dimenion of the elocity a the ratio of the dimenion of the temporal and k, indicating the reality of the wae-induced motion proce. patial frequencie, Uing the definition of the group elocity a a deriatie of the frequency with repect to the wae d number, we write imple tranformation of the expreion for the frequency : k d k. () E According to the quantum mechanic concept, we hae. A the energy Е i the kinetic energy m p of particle motion, E, where m i the particle ma and p m p i the particle m momentum, Eq. () yield k de k d dp from whence and p k, where p k p m m dp k i the de roglie wae length and k i the de roglie wae number: h p, p k. (3) According to Eq. (3), the length of the de roglie wae induced by particle motion i determined by the ratio of Planck contant to the momentum. The kinetic energy of the moing particle, a well a it momentum, it relatie in the oberer coordinate ytem. Corpucular-wae dualim i a fundamental property of moing matter; it unierality follow from the quantum concept and adequate preentation of corpucle motion in the Fourier-conjugate International Journal of Adanced Reearch in Phyical Science (IJARPS) Page 8

4 Yu.N. Dubnihche & T.Ya. Dubnihchea coordinate and frequency pace. It i related to the correlation ambiguitie, which hae a unieral character and follow from the uncertainty principle for the function t and decribing the moing particle, which are conidered a the Fourier-conjugate ignal in the coordinate and frequency pace. Thee function bear information about the moing particle and atify the uncertainty principle for the ignal t t dt d E, where Е i the ignal energy; t t dt i the energy of the econd deriatie of the Fourier pectrum of the ignal by frequency; d i the energy of the econd deriatie ignal by time. The uncertainty principle yield the uncertainty ratio t, (4) where t and are the ignal length in the coordinate pace and the width of it Fourier pectrum, repectiely: t t t E dt, E d. The function that decribe quantum mechanic object may be conidered a ignal in the coordinate and frequency pace becaue the frequency pace i an analog of the momentum pace. Multiplying Eq. (4) by Planck contant, we obtain t E, (5) where E. Thi i the ratio of Heienberg ambiguitie for fluctuation of time interal and energy at quantum mechanic cale. The momentum and frequency pace in quantum mechanic are related ap k. Multiplying and diiding the left-hand ide of inequality (4) by the group elocity, e.g., in the z direction, and taking into account that t z and k, we obtain z k, (6) where k. Expreion (6) relate the uncertaintie of the patial z and frequency k coordinate. Multiplying inequality (6) by Planck contant and taking into account that k p, we obtain Heienberg correlation ambiguitie for a quantum mechanic object in the coordinate and momentum pace: z p. (7) Thi reult in the ame way follow from the principle of uncertainty, and the uncertainty relation for the patial ignal z k when multiplying thi inequality on. Where k - the uncertainty of International Journal of Adanced Reearch in Phyical Science (IJARPS) Page 9

5 Fourier-Conjugate Model in the Corpucular-Wae Dualim Concept the patial frequency in the choen direction, z - the uncertainty of patial coordinate. In quantum mechanic, the minimum uncertainty z p occur when a quantum particle i modeled by a Gauian wae packet enelope. In thi cae the analogy trace with a minimum of uncertainty for the time-dependent t and patial z k Gauian ignal reulting from the uncertainty principle [3, 4]. There i no contradiction in corpucular-wae dualim; it jut mean that an adequate method of decription i determined by the choen method of oberation [4]. The patial-frequency image of a moing object a the Fourier pectrum of a phyical ignal i feaible. It wae propertie are manifeted, for example, in diffraction phenomena and in the Doppler effect.. CONCLUSION Wae-particle dualim i baed on the ynthei of phyical image, analogie and diplayed on language of mathematic. The concept of wae-particle duality can be decribed in term of the Fourier conjugate mathematical model of moement particle. Thee model are adequately diplayed a ignal in the coordinate pace and in the frequency (pule) pace with regard to the quantum of action. The ignal detected by an oberer in the coordinate and in the momentum pace atify to the fundamental principle of uncertainty. It follow from the uncertainty relation for the extent of the ignal in the coordinate pace and the width of it Fourier pectrum, a pecial cae of which the Heienberg uncertainty relation i. REFERENCES [] L. de roglie, enefit and leon of the hitory of cience, in: Following the Path of Science, Inotr. Lit., Mocow, 96, pp [] Papouli A., Sytem and Tranform with Application in Optic. New York: McGraw-Hill, 968. [3] Yu.N. Dubnihche, Theory and Tranformation of Signal in Optical Sytem. St. Peterburg, Lan, 0. [4] Pohl R., Optik und Atomphyik. erlin: Springer-Verlag, 963. AUTHORS IOGRAPHY Yuriy Dubnihche, Doctor of technical cience, Profeor. Intitute of Thermophyic, Siberian ranch, Ruian Academy of Science Chief reearcher. Hi current reearch interet include optical information technologie. Tatiana Dubnihchea, Doctor of phyicomathematical cience, Profeor. Nooibirk State Unierity of Economic and Management. Chair of modern natural cience and high technologie, Head of the Chair. International Journal of Adanced Reearch in Phyical Science (IJARPS) Page 30